Solve for $x$ : $ 4|x + 9| - 1 = 2|x + 9| + 3 $
Subtract $ {2|x + 9|} $ from both sides: $ \begin{eqnarray} 4|x + 9| - 1 &=& 2|x + 9| + 3 \\ \\ { - 2|x + 9|} && { - 2|x + 9|} \\ \\ 2|x + 9| - 1 &=& 3 \end{eqnarray} $ Add ${1}$ to both sides: $ \begin{eqnarray} 2|x + 9| - 1 &=& 3 \\ \\ { + 1} &=& { + 1} \\ \\ 2|x + 9| &=& 4 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x + 9|} {{2}} = \dfrac{4} {{2}} $ Simplify: $ |x + 9| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -2 $ or $ x + 9 = 2 $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -2 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -2 \\ \\ {- 9} && {- 9} \\ \\ x &=& -2 - 9 \end{eqnarray} $ $ x = -11 $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = 2 $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& 2 \\ \\ {- 9} && {- 9} \\ \\ x &=& 2 - 9 \end{eqnarray} $ $ x = -7 $ Thus, the correct answer is $x = -11 $ or $x = -7 $.